Low degree tests play an important role in classical complexity theory, serving as basic ingredients in foundational results such as $\mathsf{MIP} = \mathsf{NEXP}$ [BFL91] and the PCP theorem [AS98,ALM+98]. Over the last ten years, versions of these tests which are sound against quantum provers have found increasing applications to the study of nonlocal games and the complexity class~$\mathsf{MIP}^*$. The culmination of this line of work is the result $\mathsf{MIP}^* = \mathsf{RE}$ [JNV+20]. One of the key ingredients in the first reported proof of $\mathsf{MIP}^* = \mathsf{RE}$ is a two-prover variant of the low degree test, initially shown to be sound against multiple quantum provers in [Vid16]. Unfortunately a mistake was recently discovered in the latter result, invalidating the main result of [Vid16] as well as its use in subsequent works, including [JNV+20]. We analyze a variant of the low degree test called the low individual degree test. Our main result is that the two-player version of this test is sound against quantum provers. This soundness result is sufficient to re-derive several bounds on~$\mathsf{MIP}^*$ that relied on [Vid16], including $\mathsf{MIP}^* = \mathsf{RE}$.

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经典低个体度测试的量子稳健性

低度检验在经典复杂性理论中发挥着重要作用，作为 $\mathsf{MIP} = \mathsf{NEXP}$ [BFL91] 和 PCP 定理 [AS98,ALM+98] 等基础结果的基本成分。在过去的十年中，这些与量子证明者相抗衡的测试版本越来越多地应用于研究非局部博弈和复杂性类~$\mathsf{MIP}^*$。这一系列工作的高潮是结果 $\mathsf{MIP}^* = \mathsf{RE}$ [JNV+20]。$\mathsf{MIP}^* = \mathsf{RE}$ 的第一个报告证明中的关键成分之一是低度测试的两个证明者变体，最初在 [Vid16 ]。不幸的是，最近在后一个结果中发现了一个错误，使 [Vid16] 的主要结果及其在后续工作中的使用无效，包括 [JNV+20]。我们分析了一种称为低个体度测试的低度测试的变体。我们的主要结果是，这个测试的两人版本对量子证明者来说是合理的。这个稳健性结果足以重新推导依赖于 [Vid16] 的 ~$\mathsf{MIP}^*$ 的几个边界，包括 $\mathsf{MIP}^* = \mathsf{RE}$。